Division

Elementary division

Division is the fourth and last basic operation of arithmetic discussed in this book. Division of natural numbers involves dividing a total into a number of equal copies and count the value of a single copy. For instance, a total of 15 consist of 3 copies of 5, thus, when 15 is divided into 3 copies, each copy contains 5 (see figure 1).

Divided rectangle 3 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

Fig.1. A rectangle of 3 by 5 is 15 unit squares, divided into 3 rows (copies). Each row contains 5 unit squares: 15 ÷ 3 = 5. How would you represent 15 divided by 5 in the above picture?

Verbally 15 ÷ 3 = 5 is expressed as: "fifteen divided by three equals five" or sometimes "fifteen over three is five". Various common division signs can be used: an obelus (15 ÷ 3 = 5), a colon (15 : 3 = 5), a fraction bar ( 15 3 = 5 ) or a forward slash (15/3 = 5).

Fifteen divided by three may represent something like: "when 15 apples are equally divided between 3 persons, each person gets 5 apples". The result of a division, the quotient, can be explained as a quantity per unit. Fifteen apples equally divided over three persons results in 5 apples per person, that is, 5 apples for each person. In 15 ÷ 3 = 5 15 is called the dividend, 3 is called the divisor and 5 is called the quotient.

Division can be seen as the opposite or inverse operation to multiplication. In other words:

15 ÷ 3 = 5, because 5 × 3 = 15.

And likewise:

15 ÷ 5 = 3, because 3 × 5 = 15.

This means that the times table also reflects division. For 10 ÷ 5 = 2 we look up 10 in the table field with 5 in a corresponding cell in the top row or left column, and then the answer 2 is in the other corresponding left column or top row cell.

Division (like subtraction) is not commutative, meaning that changing the order of the division changes the result:

4 ÷ 2 ≠ 2 ÷ 4

Also note:

a ÷ a = 1, because 1 × a = a   (if a ≠ 0).

a ÷ 1 = a, because a × 1 = a.

Can you explain both by dividing apples equally between people?

Zero

Let a represent any given number, except zero. Then:

0 ÷ a = 0, because 0 × a = 0.

Dividing zero by any number other than zero always yields zero, because zero times that number always equals zero. What about dividing a (non-zero) number by zero?

5 ÷ 0 = ?, because ? × 0 = 5.

a ÷ 0 = ?, because ? × 0 = a.

We have nothing to define a ÷ 0 by, because anything times 0 equals 0. Dividing a number by zero is left undefined. Another way of looking at it is that equally dividing 5 apples between nobody is rather meaningless.

But what if a = 0? We could say: 0 ÷ 0 = 0, because 0 × 0 = 0. But anything, not only zero, multiplied by zero is zero! So, we could indeed argue that 0 ÷ 0 = 0, but we might as well argue that 0 ÷ 0 = 666 or whatever other number. Thus, 0 ÷ 0 is indeterminate and therefore also needs to be left undefined. Divide a number by zero on your calculator. What happened?

So we say:

Dividing by zero is undefined.

Euclidean division

By now you may be wondering what happens if a number does not exactly contain a number of copies, for instance 16 ÷ 3. There is no natural number that multiplied by 3 equals 16. If we divide 16 apples between 3 people, every person gets 5 apples and one apple remains undivided, unless we cut the remaining apple in 3 equal parts and give every person a piece. That last option involves fractions, which will be explained in a later chapter.

For now, we will concentrate on Euclidean division. Euclidean division only involves integers. Fractions are not included in this context. Euclidean division is also called division with remainder:

16 ÷ 3 = 5 remainder 1, because (5 × 3) + 1 = 16.

14 ÷ 3 = 4r2, because (4 × 3) + 2 = 14.

14 ÷ 4 = 3r2, because (3 × 4) + 2 = 14.

In 16 ÷ 3 = 5r1, 5 is called the quotient and 1 the remainder.

In case the remainder is zero, e.g. in 15 ÷ 3 = 5, it is said that "3 (evenly) divides 15", or that "15 is divisible by 3", or that "3 is a factor of 15", or that "15 is a multiple of 3".

Suppose we have 4 apples to equally divide between 5 people. We have not enough apples to do so, unless we start cutting and distribute only pieces of apples. But nobody will get a whole apple, so the quotient is zero and the remainder is 4.

4 ÷ 5 = 0r4, because (0 × 5) + 4 = 4.

By the way, Euclidean division also declares division by zero to be undefined, although 5 ÷ 0 = 0r5 may seem to make sense.

The long division algorithm

Let's have a look at two transcriptions of the possible thoughts of a person doing a division:

  1. 15 ÷ 4 = ?
  2. How many fours go into 15? That's 3 maximum, because 3 × 4 = 12 (and 4 × 4 = 16).
  3. The remainder is 15 − 12 = 3.
  4. Thus: 15 ÷ 4 = 3r3.

And:

  1. 153 ÷ 4 = ?
  2. 15 ÷ 4 = 3r3, because (3 × 4) + 3 = 15.
  3. 150 ÷ 4 = 30r30, because (30 × 4) + 30 = 150.
  4. So, 153 ÷ 4 = 30r(30+3).
  5. But the remainder 33 is greater than 4. It can still be divided by 4:
    33 ÷ 4 = 8r1.
  6. Thus: 153 ÷ 4 = 38r1.

The above procedure may seem a little cumbersome, but this is exactly the procedure that is methodically structured in the long division algorithm. This famous algorithm works for all divisions.

153 ÷ 4 = 38r1:

(a)quotient
(b)153 ÷ 4
(c) ... (g)algorithm steps
(h)remainder

The above example shows long division to calculate 153 ÷ 4 = 38r1. The quotient is in the top row (a) and the remainder in the bottom row (h). Again the columns from right to left represent the ones in the first column, the tens in the second column, the hundreds in the third column etc. The second row from the top (b) says "4 ) 153" which represents "153 divided by 4". Unlike the other algorithms we discussed before, we start in the left most column, the hundreds column in this example.

The recipe:
We start in row (b) with the 1 in 153 and calculate 1 ÷ 4. We write the quotient, which is zero, in the top row (a) in the hundreds place. Then in row (c) we write the result of 0 × 4. In the next row (d) we write the result of 1 − 0, which is the remainder of 1 ÷ 4. Thus: We calculate 1 ÷ 4 and write the quotient in the top row and the remiander in row (d). But the actual remainder is not 1, but 100, plus 5 in the tens place and 3 in the ones place. We bring down the 5 in the tens place, in row (d). Next we divide 15 by 4. How many times goes 4 into 15? That's 3 times, because 3 × 4 = 12. We write the 3 in the top row (a) in the tens place and the 12 in row (e). Then we write 3 (15 − 12, the remainder of 15 ÷ 4) in row (f), together with the 3 that we bring down in the ones place. How many times goes 4 into 33? That's 8 times, because 8 × 4 = 32. We write the 8 in the top row (a) in the ones place and the 32 in row (g). Finally we write 1 (33 − 32, the remainder of 33 ÷ 4) in row (h). This last remainder is also the remainder of the complete division, because there are no more digits to bring down. Thus: 153 ÷ 4 = 38r1.

In this example we could have skipped the first step with quotient zero (1 ÷ 4) and immediately moved to 15 ÷ 4:

In words:

  1. 4 goes 0 times into 1; include the next digit (5).
  2. 4 goes 3 times into 15, because 3 × 4 = 12.
  3. 15 − 12 = 3 and bring the next digit (3) down.
  4. 4 goes 8 times in 33, because 8 × 4 = 32.
  5. 33 − 32 = 1.

An other example, 445 ÷ 5 = 89:

In words:

  1. 5 goes 0 times into 4; include the next digit (4).
  2. 5 goes 8 times into 44, because 8 × 5 = 40.
  3. 44 − 40 = 4 and bring the next digit (5) down.
  4. 5 goes 9 times in 45, because 9 × 5 = 45.
  5. 45 − 45 = 0.

In the above example 5 goes 8 times into 44, but what would have happened in the long division if you would have made a mistake and concluded: "5 goes 7 times into 44" or "5 goes 9 times into 44"?

Two more examples, 6250 ÷ 25 = 250 and 13610 ÷ 15 = 907r5:

As you may have experienced by now, division by only using pen and paper (and no calculator) may take quite some calculations. When performing a long division, you may also need to use the long multiplication algorithm a number of times. It is also recommendable to check your answer. How would you do this (without a calculator) in each of the above examples?

Practice application

With the next app you can practice long division. Use the algorithm on a piece of paper to divide the numbers. Click the exercise or click "CHECK" to see if you did it right.

Exercises: